Lc coil

ABSTRACT

A coil for a magnetic resonance imaging machine includes two adjacent regions carrying currents that differ at the interface.

BACKGROUND

In magnetic resonance imaging, the rate of data acquisition is limited by how rapidly fields can be changed within the field of view. However, bounds are generally placed on how rapidly fields can be changed within an often larger region, such as a patient. With reduced fields outside the field of view, data acquisition can be acclerated.

SUMMARY

This invention provides a coil for a magnetic resonance imaging machine with two adjacent regions carrying different currents at the interface. Currents in the two regions at the interface can be in opposite directions.

The interface separating the two adjacent regions can be planar and the regions can be mirror images of each other across the interface. Current in one region and the opposite of current in the other region can be mirror images of each other across the planar interface. The current density, or volume current density integrated over the thickness of the coil, can be constant in each region.

The two adjacent regions of the coil can pass directly under and conform to a support surface, which can be flat. The cross-section of the coil can contain an arc of a circle.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows the first embodiment of the coil, side view (FIG. 1A) and axial view (FIG. 1B).

FIG. 2 shows the second embodiment of the coil, side view (FIG. 2A) and axial view (FIG. 2B).

DETAILED DESCRIPTION

FIG. 1 shows a coil within a magnetic resonance imaging machine 10. The coil has region 3 carrying current 5 and adjacent region 4 carrying current 6. Currents 5 and 6 differ at the interface 7 between the regions 3 and 4. The coil has a flat lower part 8 passing under a support surface 9 of the magnetic resonance imaging machine 10 and a partial cylindrical upper part 11 with axis 12.

Throughout this specification, the term “LC coil” refers to this invention, the terms “axial”, “z direction”, and “along z ” refer to the direction of a specified axis of the LC coil, and “LC_(z) coil” refers to an LC coil designed to produce an axial gradient of an axial magnetic field. Terms “LC_(x) coil” and “LC_(y) coil” refer to LC coils designed to produce gradiy ents of an axial magnetic field orthogonal to the axis. The term “scanner” refers to a magnetic resonance imaging machine.

1 First Embodiment

The first embodiment (FIG. 1) of an LC_(z) coil has region 3 carrying current 5 and adjacent region 4 carrying current 6. Currents 5 and 6 differ at the interface 7 between the regions 3 and 4. The first embodiment has a flat lower part S₁ 8 passing a distance Δ_(y) under a flat support surface 9 of a scanner 10 and upper part S₂ 11 a partial cylinder of radius R , angle (1+2ε_(φ))π, ε_(φ)∈[0, ½], and axis 12 coinciding with the scanner axis 13. The axis 12 is also called the coil axis. The length of the first embodiment along its axis 12 is l_(z) and the dimension of S₁ 8 are l_(x)×l_(z), with l_(x≦2). The distance between S₁ 8 and the axis 12 is y_(c) . A surface detection coil 14 is located between S₁ 8 and the support surface 9.

1.1 Definition of Coordinates

Define rectangular coordinates (x, y, z) with x parallel the support surface 9, y perpendicular to the support surface 9, and z along the coil axis 12. The flat section S₁ 8 is in the plane y=−y_(c) and is parallel to the support surface 9 in the plane y=−y_(c)+Δ_(y). The plane x=0 perpendicular to the support surface 9 contains the coil axis 12 given by the line x=0 and y=0. Centers of fields of view are arranged to lie within the plane z=0.

Cylindrical coordinates (r, φ, z) are related to coordinates (x, y, z) by the transformation x=r cos φ  (1a) y=r sin φ  (1b) The inverse transformation is $\begin{matrix} {r = \sqrt{x^{2} + y^{2}}} & \left( {2a} \right) \\ {\varphi = {{\cos^{- 1}\frac{x}{r}} = {\sin^{- 1}\frac{y}{r}}}} & \left( {2b} \right) \end{matrix}$ The partial cylindrical section S₂ 11 is at r=R and covers the angular range φ∈I _(φ)=[−ε_(φ)π, (1ε_(φ))π]  (3)

The coil dimensions l _(x)=2R cos(ε_(φ)π)  (4a) and y _(c) R sin (ε_(φ)π  (4b) 1.2 Current Density

The coil carries a current density $\begin{matrix} {{\overset{\rightarrow}{J}\left( \overset{\rightarrow}{r} \right)} = \left\{ \begin{matrix} {{{J(z)}\hat{x}},} & {\overset{\rightarrow}{r} \in S_{1}} \\ {{{J(z)}\hat{\varphi}},} & {\overset{\rightarrow}{r} \in S_{2}} \end{matrix} \right.} & (5) \end{matrix}$ where J(z) is the current density profile and {right arrow over (r)}=(x, y, z)  (6) 1.3 Field

Using the Biot-Savart law, the total field $\begin{matrix} {{{B\left( {x,y,z} \right)} = {\int_{{- l_{z}}/2}^{l_{z}/2}\quad{{\mathbb{d}z^{\prime}}{J\left( z^{\prime} \right)}{g\left( {x,y,{z - z^{\prime}}} \right)}}}}{with}} & (7) \\ \begin{matrix} {{g\left( {x,y,z} \right)} = {{\frac{\mu_{0}\left( {y + y_{c}} \right)}{4\pi}{\int_{{- l_{x}}/2}^{l_{x}/2}\quad{{\mathbb{d}x^{\prime}}\frac{1}{\left\lbrack {\left( {x - x^{\prime}} \right)^{2} + \left( {y + y_{c}} \right)^{2} + z^{2}} \right\rbrack^{3/2}}}}} +}} \\ {\frac{\mu_{0}R}{4\pi}{\int_{I_{\varphi}}\quad{{\mathbb{d}\varphi^{\prime}}\frac{R - {r\quad{\cos\left( {\varphi - \varphi^{\prime}} \right)}}}{\left\lbrack {R^{2} + r^{2} - {2{Rr}\quad{\cos\left( {\varphi - \varphi^{\prime}} \right)}} + z^{2}} \right\rbrack^{3/2}}}}} \end{matrix} & (8) \end{matrix}$ Under conditions $\begin{matrix} {{x},{{{y + y_{c}}}{{\operatorname{<<}l_{x}}/2}}} & \left( {9a} \right) \\ {{{y + y_{c}}}{{\operatorname{<<}l_{z}}/2}} & \left( {9b} \right) \\ {\sqrt{x^{2} + y^{2}}{\operatorname{<<}R}} & \left( {9c} \right) \end{matrix}$ the function $\begin{matrix} \begin{matrix} {{g\left( {x,y,z} \right)} \approx {{\frac{\mu_{0}}{2\pi}\frac{y + y_{c}}{{2{\pi\left( {y + y_{c}} \right)}^{2}} + z^{2}}} +}} \\ {\frac{\mu_{0}}{4\pi}{\frac{R^{2}}{\left\lbrack {R^{2} + z^{2}} \right\rbrack^{3/2}}\left\lbrack {{\left( {1 + {2ɛ_{\varphi}}} \right)\pi} +} \right.}} \\ \left. {\frac{l_{x}y\sqrt{x^{2} + y^{2}}}{R^{3}}\left( {\frac{3R^{2}}{R^{2} + z^{2}} - 1} \right)} \right\rbrack \end{matrix} & (10) \end{matrix}$

If the projection of the field of view onto the x−y plane is a rectangle L_(x)×L_(y) centered about (x₀, y₀) under conditions L _(x)/2, |x ₀ |<<R  (11a) L _(y)/2, |y ₀ |<<R, l _(z)/2  (11b) and y ₀ ≦−y _(c)+Δ_(y) +L _(y)/2  (11c) and the coil parameter y _(c) ≦≦R, l _(z)/2  (12) then conditions (9) are satisfied within the field of view. 1.4 Field with (14)

Define the function sgn by $\begin{matrix} {{{sgn}\quad z} = \left\{ \begin{matrix} {1,{z \geq 0}} \\ {{- 1},{x < 0}} \end{matrix} \right.} & (13) \end{matrix}$ Under conditions (9) with J(z)=J ₀ sgn z  (14) the field $\begin{matrix} {\begin{matrix} {{B\left( {x,y,z} \right)} \approx {{b\left( {x,y,z} \right)} - {\frac{1}{2}\left\lbrack {{b\left( {x,y,{z - {l_{z}/2}}} \right)} +} \right.}}} \\ \left. {b\left( {x,y,{z + {l_{z}/2}}} \right)} \right\rbrack \end{matrix}{with}} & \left( {15a} \right) \\ \begin{matrix} {{b\left( {x,y,z} \right)} = {{\frac{\mu_{0}J_{0}}{\pi}{Tan}^{- 1}\frac{z}{y + y_{c}}} +}} \\ {\frac{\mu_{0}J_{0}z}{2\sqrt{R^{2} + z^{2}}}\left\lbrack {\left( {1 + {2ɛ_{\varphi}}} \right) +} \right.} \\ \left. {\frac{l_{x}y\sqrt{x^{2} + y^{2}}}{\pi\quad R}\left( {\frac{1}{R^{2}} + \frac{1}{R^{2} + z^{2}}} \right)} \right\rbrack \end{matrix} & \left( {15b} \right) \end{matrix}$ using (2a).

Under conditions (9) and |z|≦≦l _(z)/2, R  (16) with (14), the field $\begin{matrix} {{B\left( {x,y,z} \right)} \approx {\frac{\mu_{0}J_{0}}{\pi}{Tan}^{- 1}\frac{z}{y + y_{c}}}} & (17) \end{matrix}$ the gradient $\begin{matrix} {{G_{z} = \frac{\partial B}{\partial z}}}_{{({x,y,z})} = {({x_{0},y_{0},0})}} & \left( {18a} \right) \\ {\quad{\approx \frac{\mu_{0}J_{0}}{\pi\left( {y_{0} + y_{c}} \right)}}} & \left( {18b} \right) \end{matrix}$

2 Second Embodiment

The second embodiment (FIG. 2) of an LC_(z) coil has region carrying current 17 and adjacent region 16 carrying current 18. Currents 17 and 18 differ at the interface 19 between the regions 15 and 16. The second embodiment has a flat upper part S₁ 20 passing a distance Δ_(y) under a flat support surface 21 of a scanner 22 and lower part S₂ 23 a partial cylinder of radius R , angle (1+2ε_(φ))π, ε_(φ)∈[0, ½], and axis 24 coinciding with the scanner axis 25. The axis 25 is also called the coil axis. The length of the second embodiment along its axis 24 is l_(z) and the dimension of S₁ 20 are l_(x)×l_(z), with l_(x)≦2R . The distance between S₁ 20 and the axis 24 is y_(c). A surface detection coil 26 is located between S₁ 20 and the support surface 21

2.1 Definition of Coordinates

Define rectangular coordinates (x, y, z) with x parallel the support surface 21, y perpendicular to the support surface 21, and z along the coil axis 24. The flat section S₁ 20 is in the plane y=−y_(c) and is parallel to the support surface 21 in the plane y=−y_(c)+Δ_(y). The plane x=0 perpendicular to the support surface 21 contains the coil axis 24 given by the line x=0 and y=0. Centers of fields of view are arranged to lie within the plane z=0.

Cylindrical coordinates (r, φ, z) are related to coordinates (x, y, z) by the transformation (1). The inverse transformation is (2). The partial cylindrical section S₂ 23 is at r=R and covers the angular range φ∈I _(φ)=[−(1−ε_(φ))π, −ε_(φ)π]  (19)

The coil dimensions l_(x) and y_(c) are given by (4a) and (4b).

2.2 Current Density

The coil carries a current density $\begin{matrix} {{\overset{\rightarrow}{J}\left( \overset{\rightharpoonup}{r} \right)} = \left\{ \begin{matrix} {{{J(z)}\hat{x}},} & {\overset{\rightarrow}{r} \in S_{1}} \\ {{{- {J(z)}}\hat{\varphi}},} & {\overset{\rightarrow}{r} \in S_{2}} \end{matrix} \right.} & (20) \end{matrix}$ where J(z) is the current density profile. 2.3 Field

The first and second embodiments have different partial cylindrical sections S₂ 11 and S₂ 23: current densities (5) and (20) for {right arrow over (r)}∈S₂ are carried in complementary angular ranges (3) and (19). Expressions for the field produced by the second embodiment can be obtained by modifying the expressions of Sec. (1.3).

The field $\begin{matrix} {{B\left( {x,y,z} \right)} = {\int_{{- l_{z}}/2}^{l_{z}/2}{{\mathbb{d}z^{\prime}}{J\left( z^{\prime} \right)}{g\left( {x,y,{z - z^{\prime}}} \right)}}}} & (21) \\ {with} & \quad \\ \begin{matrix} {{g\left( {x,y,z} \right)} = {\frac{\mu_{0}\left( {y + y_{c}} \right)}{4\quad\pi}{\int_{{- l_{x}}/2}^{l_{x}/2}{\mathbb{d}x^{\prime}}}}} \\ {\frac{1}{\left\lbrack {\left( {x - x^{\prime}} \right)^{\prime 2} + \left( {y + y_{c}} \right)^{2} + z^{2}} \right\rbrack^{3/2}} -} \\ {\frac{\mu_{0}R}{4\quad\pi}{\int_{I_{\varphi}}{{\mathbb{d}\varphi^{\prime}}\frac{R - {r\quad{\cos\left( {\varphi - \varphi^{\prime}} \right)}}}{\left\lbrack {R^{2} + r^{2} - {2\quad{Rr}\quad{\cos\left( {\varphi - \varphi^{\prime}} \right)}} + z^{2}} \right\rbrack^{3/2}}}}} \end{matrix} & (22) \end{matrix}$ r can be expressed in terms of x and y using (2a). Under conditions (9), $\begin{matrix} \begin{matrix} {{g\left( {x,y,z} \right)} \approx {{\frac{\mu_{0}}{2\quad\pi}\frac{y + y_{c}}{\left( {y + y_{c}} \right)^{2} + z^{2}}} -}} \\ {\frac{\mu_{0}}{4\quad\pi}{\frac{R^{2}}{\left\lbrack {R^{2} + z^{2}} \right\rbrack^{3/2}}\left\lbrack {{\left( {1 - {2ɛ_{\varphi}}} \right)\pi} +} \right.}} \\ \left. {\frac{l_{x}y\sqrt{x^{2} + y^{2}}}{R^{3}}\left( {\frac{3\quad R^{2}}{R^{2} + z^{2}} - 1} \right)} \right\rbrack \end{matrix} & (23) \end{matrix}$

If the projection of the field of view onto the x−y plane is a rectangle L_(x)×L_(y) centered about (x₀, y₀) under conditions (11) and the coil parameter y_(c) satisfies (12), then conditions (9) are satisfied within the field of view.

2.4 Field with (14)

Expressions for the field and gradient produced by the second embodiment can be obtained by modifying the expressions of Sec. (1.4).

Under conditions (9) with (14), the field $\begin{matrix} {\begin{matrix} {{B\left( {x,y,z} \right)} \approx {{b\left( {x,y,z} \right)} - {\frac{1}{2}\left\lbrack {{b\left( {x,y,z} \right)} - {l_{z}/2}} \right)} +}} \\ \left. {b\left( {x,y,{z + {l_{z}/2}}} \right)} \right\rbrack \end{matrix}{with}} & \left( {24a} \right) \\ \begin{matrix} {{b\left( {x,y,z} \right)} = {{\frac{\mu_{0}J_{0}}{\pi}{Tan}^{- 1}\frac{z}{y + y_{c}}} -}} \\ {\frac{\mu_{0}J_{0}z}{2\sqrt{R^{2} + z^{2}}}\left\lbrack {\left( {1 - {2ɛ_{\varphi}}} \right) +} \right.} \\ \left. {\frac{l_{x}y\sqrt{x^{2} + y^{2}}}{\pi\quad R}\left( {\frac{1}{R^{2}} + \frac{1}{R^{2} + z^{2}}} \right)} \right\rbrack \end{matrix} & \left( {24b} \right) \end{matrix}$

Under conditions (9) and (16) with (14), the field B is (17) and the gradient G_(z) is (18)

3 MRI Using an LC Coil

3.1 Field Profiles

The three types of LC coil are LC_(x), LC_(y), and LC_(z). An LC_(x) coil produces an LC_(X) field b({right arrow over (r)}, ι)=G _(x)(ι){tilde over (x)}({right arrow over (r)})  (25a) with gradient $\begin{matrix} {G_{x} = \frac{\partial B}{\partial x}} & \left( {25b} \right) \end{matrix}$ An LC_(y) coil produces an LC_(y) field B({right arrow over (r)}, t)=G _(y)(t){tilde over (y)}({right arrow over (r)})  (26a) with gradient $\begin{matrix} {G_{y} = \frac{\partial B}{\partial y}} & \left( {26\quad b} \right) \end{matrix}$

An LC_(z) coil produces an LC_(z) field B({right arrow over (r)}, ι)=G _(z)(ι){tilde over (z)}({right arrow over (r)})  (27 with gradient $\begin{matrix} {G_{z} = \frac{\partial B}{\partial z}} & \left( {27b} \right) \end{matrix}$ The gradients are evaluated at the center of the field of view.

Magenetic resonance imaging with an LC_(z) coil requires that {tilde over (z)}({right arrow over (r)}) satisfy {tilde over (z)}=0 at the center of the field of view  (28a) θ{tilde over (z)}/θz=1 at the center of the field of view  (28b) θ{tilde over (z)}/θz≧0 within the field of view  (28c) and {tilde over (z)} attains values for fixed x and y within the field of view that are unique within the region of sensitivity of the detection coil  (28d) Magnetic resonance imaging with an LC_(x) coil requires that {tilde over (x)}({right arrow over (r)}) satisfy (28) with x

z and {tilde over (x)}

{tilde over (z)}. Magnetic resonance imaging with an LC_(y) coil requires that {tilde over (y)}({right arrow over (r)}) satisfy (28) with y

z and {tilde over (y)}

{tilde over (z)}.

The center of a field of view refers to the center of an image in field coordinates. For example, if fields linear in x and y are used with an LC_(z) field linear in {tilde over (z)}, the field of view is a rectangular box in field coordinates (x, y, {tilde over (z)}) and the center of the field of view refers to the center of the box.

3.2 Imaging Parameters

This section assumes the use of an LC_(z) field together with fields linear in x and y . Similar equations hold for other combinations of LC and linear fields.

3.2.1 Non-Oblique Imaging Parameters

In field coordinates (x, y, {tilde over (z)}) , the field of view is a box L_(x)×L_(y)×{tilde over (L)}_(z) centered about (x, y, {tilde over (z)})=(x ₀ , y ₀, 0)

(x, y, z)=(x ₀ , y ₀ , z ₀)  (29) With choices for pixel numbers N_(x), N_(y), N_(z), and pixel sizes Δx, Δy, Δz , the imaging parameters are $\begin{matrix} {L_{x} = {N_{x}\Delta\quad x}} & \left( {30a} \right) \\ {L_{y} = {N_{y}\Delta\quad y}} & \left( {30b} \right) \\ {{\overset{\sim}{L}}_{z} = {N_{z}\Delta\quad z}} & \left( {30c} \right) \\ {k_{x} = {n_{x}\Delta\quad k_{x}}} & \left( {31a} \right) \\ {k_{y} = {n_{y}\Delta\quad k_{y}}} & \left( {31b} \right) \\ {k_{z} = {n_{z}\Delta\quad k_{z}}} & \left( {31c} \right) \\ {n_{x} \in \left\{ {{{- N_{x}}/2},\ldots\quad,{{N_{x}/2} - 1}} \right\}} & \left( {32a} \right) \\ {n_{y} \in \left\{ {{{- N_{y}}/2},\ldots\quad,{{N_{y}/2} - 1}} \right\}} & \left( {32b} \right) \\ {n_{z} \in \left\{ {{{- N_{z}}/2},\ldots\quad,{{N_{z}/2} - 1}} \right\}} & \left( {32c} \right) \\ {{\Delta\quad k_{x}} = \frac{2\pi}{L_{x}}} & \left( {33a} \right) \\ {{\Delta\quad k_{y}} = \frac{2\quad\pi}{L_{y}}} & \left( {33b} \right) \\ {{\Delta\quad k_{z}} = \frac{2\pi}{{\overset{\sim}{L}}_{z}}} & \left( {33c} \right) \end{matrix}$

Fourier reconstruction yields an image on a grid in field coordinates: (x, y, {tilde over (z)})=(n _(x) Δx+x ₀ , n _(y) Δy+y ₀ , n _(z) Δz)  (34) The image can be scaled to coordinates (x, y, z) using the 1-to-1 mapping (x, y, z)

(x, y, {tilde over (z)}) within F∩D  (35) The resolutions in coordinates (x, y, z) are accurately given by (Δx, Δy, Δz)J′=(Δx, Δy, Δz(θ{tilde over (z)}/θz) ⁻¹)  (36) where Jacobian $\begin{matrix} {J^{\prime} = {\frac{\partial\left( {x,y,z} \right)}{\partial\left( {x,y,\overset{\sim}{z}} \right)} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \left( {{\partial\overset{\sim}{z}}/{\partial z}} \right)^{- 1} \end{bmatrix}}} & (37) \end{matrix}$ using the relation θz/θ{tilde over (z)}=(θ{tilde over (z)}/θz)⁻¹  (38) which follows from the fact that both θz/θ{tilde over (z)}and θ{tilde over (z)}/θz are taken at constant x and y . At the center of the field of view (29), the Jacobian J′is the identity matrix and the resolutions are Δx, Δy, Δz . 3.2.2 Double-Oblique Imaging Parameters

Coordinates (x′, y′, z′) and field coordinates ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′) are obtained from coordinates (x, y, z) and field coordinates (x, y, {tilde over (z)}) by a rotation by θ₁ about z and a rotation by θ₂ about x : (x′, y′, z′)=(x, y, z)R

(x, y, z)=(x′, y′, z′)R ⁻¹  (39) and ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′)=(x, y, {tilde over (z)})R

(x, y, {tilde over (z)})=({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′)R ⁻¹  (40) where the rotation matrix $\begin{matrix} {R = \begin{bmatrix} {\cos\quad\theta_{1}} & {{- \sin}\quad\theta_{1}} & 0 \\ {\sin\quad\theta_{1}\quad\cos\quad\theta_{2}} & {\cos\quad\theta_{1}\quad\cos\quad\theta_{2}} & {{- \sin}\quad\theta_{2}} \\ {\sin\quad\theta_{1}\quad\sin\quad\theta_{2}} & {\cos\quad\theta_{1}\quad\sin\quad\theta_{2}} & {\cos\quad\theta_{2}} \end{bmatrix}} & (41) \end{matrix}$ and inverse matrix $\begin{matrix} {R^{- 1} = \begin{bmatrix} {\cos\quad\theta_{1}} & {\sin\quad\theta_{1}\quad\cos\quad\theta_{2}} & {\sin\quad\theta_{1}\quad\sin\quad\theta_{2}} \\ {{- \sin}\quad\theta_{1}} & {\cos\quad\theta_{1}\quad\cos\quad\theta_{2}} & {\cos\quad\theta_{1}\quad\sin\quad\theta_{2}} \\ 0 & {{- \sin}\quad\theta_{2}} & {\cos\quad\theta_{2}} \end{bmatrix}} & (42) \end{matrix}$

Consider combining fields linear in x , y , and {tilde over (z)} according to (40) to create fields linear in {tilde over (x)}′, {tilde over (y)}′, and {tilde over (z)}′. In field coordinates ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′) , the field of view is a box {tilde over (L)}_(x′)×_(y′)×{tilde over (L)}_(z′) centered about $\begin{matrix} \begin{matrix} {\left( {x,y,\overset{\sim}{z}} \right) = \left( {x_{0},y_{0},0} \right)} & \Leftrightarrow & {\left( {{\overset{\sim}{x}}^{\prime},{\overset{\sim}{y}}^{\prime},{\overset{\sim}{z}}^{\prime}} \right) = \left( {{\overset{\sim}{x}}_{0}^{\prime},{\overset{\sim}{y}}_{0}^{\prime},{\overset{\sim}{z}}_{0}^{\prime}} \right)} \\ \quad & \Leftrightarrow & {\left( {x^{\prime},y^{\prime},z^{\prime}} \right) = \left( {x_{0}^{\prime},y_{0}^{\prime},z_{0}^{\prime}} \right)} \end{matrix} & (43) \end{matrix}$ With choices for pixel numbers N_(x′), N_(y′), N_(z′), and pixel numbers Δx′, Δy′, Δz′, the imaging parameters are {tilde over (L)} _(x′) =N _(x′) Δx′  (44a) {tilde over (L)} _(y′) =N _(y′) Δy′  (44b) {tilde over (l)} _(z′) =N _(z′) Δz′  (44c) $\begin{matrix} {k_{x^{\prime}} = {n_{x^{\prime}}\Delta\quad k_{x^{\prime}}}} & \left( {45a} \right) \\ {k_{y^{\prime}} = {n_{y^{\prime}}\Delta\quad k_{y^{\prime}}}} & \left( {45b} \right) \\ {k_{z^{\prime}} = {n_{z^{\prime}}\Delta\quad k_{z^{\prime}}}} & \left( {45c} \right) \\ {n_{x^{\prime}} \in \left\{ {{{- N_{x^{\prime}}}/2},\ldots\quad,{{N_{x^{\prime}}/2} - 1}} \right\}} & \left( {46a} \right) \\ {n_{y^{\prime}} \in \left\{ {{{- N_{y^{\prime}}}/2},\ldots\quad,{{N_{y^{\prime}}/2} - 1}} \right\}} & \left( {46b} \right) \\ {n_{z^{\prime}} \in \left\{ {{{- N_{z^{\prime}}}/2},\ldots\quad,{{N_{z^{\prime}}/2} - 1}} \right\}} & \left( {46c} \right) \\ {{\Delta\quad k_{x^{\prime}}} = \frac{2\pi}{{\overset{\sim}{L}}_{x^{\prime}}}} & \left( {47a} \right) \\ {{\Delta\quad k_{y^{\prime}}} = \frac{2\quad\pi}{{\overset{\sim}{L}}_{y^{\prime}}}} & \left( {47b} \right) \\ {{\Delta\quad k_{z^{\prime}}} = \frac{2\pi}{{\overset{\sim}{L}}_{z^{\prime}}}} & \left( {47c} \right) \end{matrix}$

Fourier reconstruction yields an image on a grid: ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)})=(n _(x′) Δx′+{tilde over (x)}′ ₀ , n _(y′) Δy′+{tilde over (y)} ₀ , n _(z′) Δz′+{tilde over (z)} ₀)  (48) The image can be scaled to coordinates (x′, y′, z′) using the 1-1 mapping (35) and the transformations (39) and (40): ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′)→(x, y, {tilde over (z)})→(x, y, z)→(x′, y′, z′)  (49) The resolutions in coordinates (x′, y′, z′) are accurately given by (Δx′, Δy′, Δz′)J′  (50) where Jacobian {tilde over (J)}′=R ⁻¹ J′R  (51) At the center of the field of view (43), the Jacobians J′ and {tilde over (J)}′ are the identity matrices and the resolutions Δx′, Δy′, Δz′. 3.3 Additional Parameters

The parameter κ is defined by $\begin{matrix} {\kappa = {\frac{{B}_{\max_{P}}}{{B}_{\max_{F\bigcap D\bigcap P}}} \geq 1}} & (52) \end{matrix}$ where B|_(max P) and B|_(max F∩D∩P) indicate the maximum values of B attained over a region P , such as a patient, and over the intersection F∩D∩P , where F is the field of view and D is the region of sensitivity of the detection coil. A second parameter κ_(D) is defined by $\begin{matrix} {k_{D} = {\frac{{B}_{\max_{D\bigcap P}}}{{B}_{\max_{F\bigcap D\bigcap P}}} \in \left\lbrack {1,k} \right\rbrack}} & (53) \end{matrix}$ 4 MRI Using First and Second Embodiments

Field Coordinate with (14)

The current densities of the first and second embodiments are (5) and (20). Under conditions (9) and (16) with (14), the field is (17) and field coordinate $\begin{matrix} {\overset{\sim}{z} = \frac{B}{G_{z}}} & \left( {54a} \right) \\ {\approx {\left( {y_{0} + y_{c}} \right){Tan}^{1}\frac{z}{y + y_{c}}}} & \left( {54b} \right) \end{matrix}$ Within a field of view F that is a rectangular box L_(x)×L_(y)×{tilde over (L)}_(z) in coordinates (x, y, {tilde over (z)}) centered about x₀, y₀, 0) , (9) is satisfied given (11) and (12), and (16) is satisfied given $\begin{matrix} {{{z}_{maxF} = {\left( {\frac{L_{y}}{2} + y_{0} + y_{c}} \right)\tan\frac{L_{z}}{2\left( {y_{0} + y_{c}} \right)}{{\operatorname{<<}l_{\quad z}}/2}}},R} & \left( {55a} \right) \end{matrix}$ and {tilde over (L)} _(z)≦π(y ₀ +y _(c))  (55b) The field coordinate {tilde over (z)} (54) then satisfies (28). 4.2 Improved Fields

The ideal LC_(z) field B ^(id)({right arrow over (r)}, t)=G _(z) {tilde over (z)} ^(id)(z; {tilde over (L)} _(z))  (56) where the field coordinate $\begin{matrix} {{{\overset{\sim}{z}}^{id}\left( {z;{\overset{\sim}{L}}_{z}} \right)} = \left\{ \begin{matrix} {{{\overset{\sim}{L}}_{z}/2},} & {z > {{\overset{\sim}{L}}_{z}/2}} \\ {z,} & {z \in \left\lbrack {{{- {\overset{\sim}{L}}_{z}}/2},{{\overset{\sim}{L}}_{z}/2}} \right\rbrack} \\ {{{- {\overset{\sim}{L}}_{z}}/2},} & {z < {{- {\overset{\sim}{L}}_{z}}/2}} \end{matrix} \right.} & (57) \end{matrix}$

For a field of view with {tilde over (z)}^(id)∈[−{tilde over (L)}_(z)/2{tilde over (L)}_(z)/2], the field B^(id) has the values κ=κ_(D)=1 and a uniform resolution Δz(θ{tilde over (z)}/θz)⁻¹ (36) along z for z∈[−{tilde over (L)}_(z)/2, {tilde over (L)}_(z)/2].

Curent density profiles J(z) that produce fields B better approximating the ideal fields B^(id) (56) over D than (17) can be calculated. The field B(x₀, y₀, z) is given by (7), where g(x₀, y₀, z) is (8) for the first embodiment and (22) for the second embodiment. The integral (7) can be approximated by a sum: $\begin{matrix} {B_{i} \approx {\delta\quad z{\sum\limits_{j = {- n}}^{n}\quad{g_{i - j}J_{j}}}}} & (58) \end{matrix}$ where B _(j) =B(x ₀ , y ₀ , jδz)  (59a) J _(j) =J(jδz)  (59b) g _(j) =g(x ₀ , y ₀ , jδz)  (59c) and J(z) either vanishes or is neglected for |z|≧nδz

Defining the matrix G _(ij) =δzg _(1−j)  (60) and treating B_(j) and J_(j) as column vectors, the approximation (58) becomes the matrix equation $\begin{matrix} {B_{i} \approx {\sum\limits_{j = {- n}}^{n}\quad{G_{ij}J_{j}}}} & (61) \end{matrix}$

For the first embodiment $\begin{matrix} {G_{ij}\underset{(8)}{=}{{\frac{\mu_{0}Y\quad\delta\quad z}{4\pi}{\int_{{- l_{x}}/2}^{l_{x}/2}\quad{{\mathbb{d}x^{\prime}}\frac{1}{\left\lbrack {X^{2} + Y^{2} + {\left( {\delta\quad z} \right)^{2}\left( {i - j} \right)^{2}}} \right\rbrack^{3/2}}}}} + {\frac{\mu_{0}R\quad\delta\quad z}{4\pi}{\int_{I_{\varphi}}^{\quad}\quad{{\mathbb{d}\varphi^{\prime}}\frac{R - {r_{0}\cos\quad\Phi}}{\left\lbrack {R^{2} + r_{0}^{2} - {2{Rr}_{0}\cos\quad\Phi} + {\left( {\delta\quad z} \right)^{2}\left( {i - j} \right)^{2}}} \right\rbrack^{3/2}}}}}}} & (62) \end{matrix}$ where $\begin{matrix} {X = {x_{0} = x^{\prime}}} & \left( {63a} \right) \\ {Y = {y_{0} + y_{c}}} & \left( {63b} \right) \\ {\Phi = {\varphi_{0} - \varphi^{\prime}}} & \left( {63c} \right) \\ {r_{0} = \sqrt{x_{0}^{2} + y_{0}^{2}}} & \left( {63d} \right) \\ {\varphi_{0} = {{\cos^{- 1}\frac{x_{0}}{r_{0}}} = {\sin^{- 1}\frac{y_{0}}{r_{0}}}}} & \left( {63e} \right) \end{matrix}$ Both terms are decreasing functions of i−j . In addition, the first term is positive for y₀≦−y_(c) and the second term is positive for r₀≦R , using $\begin{matrix} {0 < {R - r_{0}} \leq {R - {r_{0}\quad\cos\quad\left( {\varphi - \varphi^{\prime}} \right)}}} & \left( {64a} \right) \\ {and} & \quad \\ {{0 < \left( {R - r_{0}} \right)^{2}} = {R^{2} + r_{0}^{2} - {2R\quad r_{0}}}} & \left( {64b} \right) \\ {\quad{\leq {R^{2} + r_{0}^{2} - {2R\quad r_{0}\quad\cos\quad\left( {\varphi - \varphi^{\prime}} \right)}}}} & \quad \end{matrix}$ Therefore, G_(ij) is a positive, decreasing function of i−j for y₀≦−y_(c) and r₀≦R , and the determinant $\begin{matrix} {{\det\quad G} = {\sum\limits_{\sigma \in S_{{2n} + 1}}^{\quad}\quad{\left( {{sgn}{\quad\quad}\sigma} \right)G_{{- n},\sigma_{- n}}\ldots\quad G_{n,\sigma_{n}}}}} & (65) \end{matrix}$ can be written as an alternating sum of decreasing terms. Consequently, det G≠0 and the inverse matrix (G⁻¹)_(ji) exists.

For the second embodiment $\begin{matrix} {G_{ij}\underset{(23)}{\approx}{\frac{\mu_{0}Y\quad\delta\quad z}{2{\pi\left\lbrack {Y^{2} + {\left( {\delta\quad z} \right)^{2}\left( {i - j} \right)^{2}}} \right\rbrack}} - {\frac{\mu_{0}R^{2}\delta\quad z}{4{\pi\left\lbrack {R^{2} + {\left( {\delta\quad z} \right)^{2}\left( {i - j} \right)^{2}}} \right\rbrack}^{3/2}}\quad\left\lbrack \quad{{\left( {1 - {2ɛ_{\varphi}}} \right)\pi} + {\left( {\frac{3R^{2}}{R^{2} + {\left( {\delta\quad z} \right)^{2}\left( {i - j} \right)^{2}}} - 1} \right)\frac{l_{x}r_{0}y_{0}}{R^{3}}}} \right\rbrack}}} & (66) \end{matrix}$ under conditions (9) on (x, y)=(x₀, y₀) and using (63). Defining $\begin{matrix} {W = {{\left( {1 - {2\quad ɛ_{\varphi}}} \right)\pi} + {\frac{2l_{x}r_{0}}{R^{3}}\min\left\{ {y_{0},0} \right\}}}} & (67) \end{matrix}$ G_(ij) is a positive, decreasing function of i−j for 0≦Y≦≦R  (68) and $\begin{matrix} {{n\quad\delta\quad z} < \sqrt{\frac{{RW}_{yo}}{2}}} & (69) \end{matrix}$ and the determinant det G (65) can be written as an alternating sum of decreasing terms. Consequently, det G≠0 and the inverse matrix (G⁻¹)_(ji) exists.

Using the inverse matrix (G⁻¹)_(ji), the equation $\begin{matrix} {J_{j} \approx {\sum\limits_{i = {- n}}^{n}\quad{\left( G^{- 1} \right)_{ji}B_{i}}}} & (70) \end{matrix}$ can be used to find J_(j) required to produce specified field values B_(i) .

Let D be such that D∈{(x, y, z): |z|≦nδz}  (71) and let {tilde over (b)}(z) be a smooth function better approximating the ideal field B^(id) over D∩λthan (17), where λis the line λ={(x, y, z): x=x ₀ , y=y ₀}  (72) With B _(i) ={tilde over (b)}(iδz)  (73) values of {tilde over (b)}(z) J_(j) can be calculated from (70) and a current density profile J(z) constructed by connect-the-dots. The smooth field B(x, y, z) (7) produced by J(z) better approximates B^(id) over D than (17). 4.3 Adjustable Field of View

The first and second embodiments can be designed with several fields of view η₁ {tilde over (L)} _(z)≦η₂ {tilde over (L)} _(z)≦ . . . ≦η_(N) {tilde over (L)} _(z)  (74) in mind. Let fields G_(z){tilde over (z)}_(a), a=1, . . . , N , approximate ideal fields (57) over D : {tilde over (z)} _(a) |D≈{tilde over (z)} ^(id)(z; η _(a) {tilde over (L)} _(z))|D  (75) The notation D means restricted to D . Define fields B_(a) by B ₁ =G _(z) {tilde over (z)} _(a=1)  (76a) and, for a=2, . . . , N , B _(a) G _(z)({tilde over (z)}_(a) {tilde over (z)} _(a−1))  (76b) Current density profiles J_(a)(z) producing fields approximating B_(a) can be found by the method of Sec. (4.2). With components a=1, . . . , N carrying J_(a)(z) , the first k≦N components generate a field B ^((k)) =B ₁ + . . . +B _(k) =G _(z) {tilde over (z)} _(k)  (77a) with B ^((k)) |D≈{tilde over (z)} ^(id)(z; η _(k) {tilde over (L)} _(z))|D  (77b) 5 Advantage of LC Coil for MRI

The gradient of an LC coil with smaller fields outside the field of view can be switched more rapidly without violating a bound on the field rate of change. Consequently, larger regions of k -space can be covered within a given time.

It is to be understood that while the invention has been described in conjunction with the detailed description thereof, the foregoing description is intended to illustrate and not limit the scope of the invention, which is defined the scope of the appended claims. Other aspects, advantages, and modifications are within the scope of the following claims. 

1. An apparatus comprising: a current-carrying element with a first region carrying a first current; a second region adjacent to said first region; said second region carrying a second current; said second current differing from said first current where adjacent; and a magnetic resonance imaging machine.
 2. Said apparatus of claim 1 wherein: said second region separated from said first region by a planar interface; and said first and second regions being mirror images in said planar interface.
 3. Said apparatus of claim 1 wherein: said first current has a first direction; said second current has a second direction; and said second direction is opposite said first direction.
 4. Said apparatus of claim 2 wherein opposite of said second current being a mirror image of said first current in said planar interface.
 5. Said apparatus of claim 4 wherein: said first region carries a first volume current density; said first region has a first thickness; said first region carries a first current density, comprising said first volume current density integrated over said first thickness of said first region; and said first current density is constant over said first region.
 6. Said apparatus of claim 1 comprising a support surface.
 7. Said apparatus of claim 6 wherein said first and second regions pass under and conform to said support surface.
 8. Said apparatus of claim 7 wherein said support surface is flat.
 9. Said apparatus of claim 8 wherein said current-carrying element has cross-section comprising an arc of a circle. 